{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/pearson-s-r-adjusted","title":"Adjust Pearson's $r$ to Measure Arbitrary Monotone Dependence","arxiv_id":"2205.04571","date":"2022-05-09","proceeding":null,"authors":["Xinbo Ai"],"abstract":"Pearson's r, the most widely-used correlation coefficient, is traditionally regarded as exclusively capturing linear dependence, leading to its discouragement in contexts involving nonlinear relationships. However, recent research challenges this notion, suggesting that Pearson's r should not be ruled out a priori for measuring nonlinear monotone relationships. Pearson's r is essentially a scaled covariance, rooted in the renowned Cauchy-Schwarz Inequality. Our findings reveal that different scaling bounds yield coefficients with different capture ranges, and interestingly, tighter bounds actually expand these ranges. We derive a tighter inequality than Cauchy-Schwarz Inequality, leverage it to refine Pearson's r, and propose a new correlation coefficient, i.e., rearrangement correlation. This coefficient is able to capture arbitrary monotone relationships, both linear and nonlinear ones. It reverts to Pearson's r in linear scenarios. Simulation experiments and real-life investigations show that the rearrangement correlation is more accurate in measuring nonlinear monotone dependence than the three classical correlation coefficients, and other recently proposed dependence measures.","url_abs":"https://arxiv.org/abs/2205.04571v7","url_pdf":"https://arxiv.org/pdf/2205.04571v7.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"pearson-s-r-adjusted","repo_url":"https://github.com/byaxb/recor","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2205.04571","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}